THE ISOPERIMETRIC INEQUALITY VIA APPROXIMATION THEORY AND FREE BOUNDARY PROBLEMS

CATHERINE BEN´ ETEAU´ AND DMITRY KHAVINSON

Abstract. In this survey paper, we examine the isoperimetric inequality from an analytic point of view. We use as a point of departure the concept of analytic content in approximation theory: this approach reveals ties to overdetermined boundary problems and hydrodynamics. In particular, we look at problems connected to determining the shape of an electrified droplet or equivalently, that of an air bubble in fluid flow. We also discuss the connection with the Schwarz function and quadrature domains. Finally, we survey some known generalizations to higher dimensions and list many open problems that remain. This paper is an expanded version of the plenary talk given by the second author at the fourth CMFT conference in Joensuu, Finland, in June 2005.

1. History

The isoperimetric theorem deals with the question of finding, among all simple closed curves of a given length P , the curve that surrounds the largest area. It was already known to the Greeks that the answer is the circle of radius R = P/2π : Pappus and later Theon of Alexandria (III A.D.) already state the result and give credit to Zenodorus (see [4, 6] for a historical discussion). We thus obtain the isoperimetric inequality: (1.1) Area = A ≤ π(P/2π)2 ⇔ 4πA ≤ P 2. “Cutting a hole” in a domain will increase P and diminish A, so (1.1) holds for multiply connected domains as well. J. Steiner [58] in 1838 gave the first proof of the result assuming the solution exists. It was completed by P. Edler [15] in 1882, H.A. Schwarz [54] in 1884 and C. Carath´eodory and E. Study [8] in 1910. Schwarz also proved the three dimensional version S3 ≥ 36πV 2. E. Schmidt [53] in 1938 published a proof of the n-dimensional version: Sn ≥ 2πn/2nn−1V n−1/Γ(n/2). A survey by Bl¨asj¨o[6], giving in particular a nice account of the early geometric “proofs” of the isoperimetric theorem, has recently appeared. One of the early analytic proofs of (1.1) was given by A. Hurwitz [26] in 1901. Let us sketch his argument here.

2000 Mathematics Subject Classification. Primary: 30H05, Secondary: 35R35. The second author gratefully acknowledges support from the National Science Foundation. 1 2 C. BEN´ ETEAU´ AND D. KHAVINSON

Sketch of proof. Suppose the region Ω is bounded by the simple closed smooth curve Γ, parametrized with respect to the arc-length parameter s and with length 2π. (So, the isoperimetric inequality would state that A ≤ π.) That is, we can write X∞ ins Γ := {z(s) = cne }, −∞ where |z0(s)| = 1. Then Z 1 2π X∞ 1 = |z0(s)|2ds = n2|c |2. 2π n 0 −∞ On the other hand, using Green’s theorem, Z 1 A = (xdy − ydx) 2 Γ Z 1 = Im z¯(s)z0(s)ds 2 Γ X∞ 2 = π n|cn| . −∞

2 Since n ≤ n , A ≤ π. A = π ⇔ cn = 0, n 6= 0, 1, i.e.

ins Γ := {c0 + c1e }, a circle. 2

Another proof using the Riemann mapping theorem was given by Carleman in 1921. Theorem 1.1. ([9]) Suppose Ω is a Jordan domain with a rectifiable boundary. Then for all f analytic in Ω and continuous in the closure of Ω, ZZ Z 1 |f|2dA ≤ ( |f|ds)2. Ω 4π Γ

Notice that letting f = 1 yields the isoperimetric inequality. Also noteR that the 1 2 above inequality would be rather trivial if we had the expression 4π Γ |f| ds on the right hand side; that inequality can be obtained immediately from the similar inequality in the unit disk via conformal mapping. The Carleman inequality, however, requires an additional neat trick beyond a change of variables. In this survey paper, we are interested in the analytic approach to the isoperimet- ric inequality and its connections with approximation theory and free boundary problems. This approach reveals a close tie to hydrodynamics and, in particu- lar, to problems concerning shapes of electrified droplets of perfectly conducting fluid. We use as a point of departure the paper [31], in which the author discusses THE ISOPERIMETRIC INEQUALITY 3 the concept of analytic content, and the related survey paper [18]. In Section 3, we discuss the connection with overdetermined boundary value problems and Serrin’s theorem. In Section 4, we describe a more general problem and its appli- cation to determining the shape of a droplet of conducting fluid in the presence of an electric field. In Section 5, we examine some special cases of this more general problem in connection with the Schwarz function, Vekua’s problem, and quadrature domains. Finally, in Section 6, we survey known generalizations to higher dimensions and state many remaining open problems. Acknowledgements. The authors would like to thank Dr. Van Deursen and Dr. Darren Crowdy for pointing out several interesting references to electrowetting.

2. Analytic Content

Let Ω be a finitely connected region in C bounded by n simple closed analytic curves γj, j = 1, . . . , n. Let Γ = ∪γj be the boundary of the region Ω.AΩ denotes the set of functions analytic in Ω and continuous in Ω¯, and C(Ω)¯ is the set of continuous functions in Ω¯. Let R(Ω)¯ denote the closure (in the uniform norm) of the set of rational functions with poles outside of Ω¯. In general, not every function analytic in Ω and continuous in Ω¯ is an element of R(Ω)¯ , but it is well- known that for the finitely connected regions Ω with analytic boundaries that ¯ we are considering here, AΩ = R(Ω) (see, for example, [16]). In general, it is an interesting and central question of approximation theory to determine when a function is an element of R(Ω)¯ , or what its distance to R(Ω)¯ is. In particular, we will be interested in approximating the “simplest” non-analytic function, namely z.¯ The following definition was introduced in [30]. Definition. The analytic content of a domain Ω is

λ(Ω) := inf k z¯ − ϕ kC(Ω)¯ . ϕ∈AΩ

Therefore the analytic content of a compact set Ω¯ may be thought of as the distance between the functionz ¯ and the algebra R(Ω)¯ , or in the case of finitely connected domains, AΩ. The following theorem gives a quantitative geometric estimate of the analytic content of a set - and also yet another proof of the isoperimetric inequality! Theorem 2.1. Let Ω and Γ be as above, and let A and P be the area and perimeter of Ω, respectively. Then r 2A A ≤ λ(Ω) ≤ , P π q 2 A so P ≥ 4πA. Moreover, λ(Ω) = π ⇔ Ω = disk.

This theorem is discussed in detail in [18]. Let us outline the argument here. H. Alexander in 1973 ([3]) proved the upper estimate by noticing the connection with the Ahlfors-Beurling estimate from 1950 ([1]). 4 C. BEN´ ETEAU´ AND D. KHAVINSON

More specifically, suppose D is a bounded domain (with smooth boundary ∂D) containing Ω¯. By the Cauchy-Green formula, Z Z 1 z¯ 1 1 ζ¯ = dz − dA(z), 2πi ∂D z − ζ π D z − ζ where dA is area measure. Define Z 1 1 G(ζ) = dA(z). π Ω¯ z − ζ Then Z Z 1 z¯ 1 1 ζ¯ + G(ζ) = dz − dA(z). 2πi ∂D z − ζ π D−Ω¯ z − ζ

The right hand side can be easily seen to be in AΩ. Therefore λ ≤ max |G(ζ)|. ζ∈Ω¯ The Ahlfors-Beurling estimate shows that r A |G(ζ)| ≤ , π with equality holding only for a disk, thus proving the theorem. Gamelin and D. Khavinson gave a simple proof of the Ahlfors-Beurling estimate (see [18, p. 25- 28]). Their argument goes as follows. Without loss of generality, maxζ∈Ω¯ |G(ζ)| is attained at z = 0, and, rotating, equals Z 1 dA(ζ) Re = λmax. π Ω ζ Now notice that {ζ : Re 1 ≥ c} = {ζ : |ζ − 1 | ≤ 1 } =: ∆, a disk. For this disk, q ζ 2c 2c 1 A(∆) λ = 2c = π , while it is easy to show that, for any Ω 6= ∆, the integral is smaller, as desired. D. Khavinson proved the lower estimate in [30] by applying the complex form of Green’s theorem to the functionz ¯−ϕ(z), where ϕ ∈ AΩ is the best approximation toz. ¯ This gives Z Z (¯z − ϕ(z))dz = 2i dA(z) = 2iA, Γ Ω so that Z λP ≥ | (¯z − ϕ(z))dz| ≥ 2A. Γ Theorem 2.1 leads naturally to the following:

2A Question. For which Ω does the equality P = λ(Ω) hold?

The next theorem and following remarks are taken from [31]. THE ISOPERIMETRIC INEQUALITY 5

Theorem 2.2. ([31]) Let Ω and Γ be as above. The following are equivalent: 2A (i) λ = P ;

(ii)There is ϕ ∈ AΩ such that z¯(s) − iλz¯˙(s) = ϕ(z(s)) on Γ, where s is the arc-length parameter; R R 1 1 (iii) A Ω fdA = P Γ fds for all f ∈ AΩ. Remark. Note that (iii) holds for annuli Ω = {r < |z| < R}. Simply take the Laurent series decomposition of f = f1 + f2 in the annulus, where f1 is analytic inside {z : |z| < R} and f2(∞) = 0, and notice that both sides of the equality in (iii) are equal to f1(0). Also, (ii) easily implies that if Γ contains a circular arc, Ω is a disk or an annulus. Indeed, suppose, for simplicity, that Γ contains an arc centered at the origin and of radius R. Then we can write z(s) = Reis/R, where s is the arc-length parameter, and a calculation shows that R2 − Rλ c z¯(s) − iλz¯˙(s) = = = ϕ(z) z(s) z on that arc. But since ϕ ∈ AΩ, this equality must hold on all of Γ. If c = 0, then ϕ = 0 and |z| = const on Γ, so Γ is a disk centered at the origin. If c 6= 0, iλzz¯˙(s) = |z|2 − c, ¯ 1 d|z|2 2 or, in other words, Re(zz˙(s)) = 2 ds = 0. Therefore |z| is a constant locally on ∂Ω, so Ω is an annulus. The following conjecture remains open. Conjecture 2.1. ([31]) 2A λ = ⇔ Ω is a disk, or an annulus. P In the simply-connected case, we know more. Theorem 2.3. ([31])

2A If λ = and Ω is simply connected, Ω is a disk. P

2A Sketch of proof. Since λ = P , there exists ϕ ∈ AΩ such that (2.1)z ¯(s) − iλz¯0(s) = ϕ(z(s)). Differentiating, we obtain iλz¨ (2.2) z¯˙(1 + ) = ϕ0(z)z. ˙ z˙ Sincez, ˙ z¨ are orthogonal vectors, we can rewrite (2.2) as iλz¨ (2.3) Re(1 + ) = ϕ0(z)(z ˙)2. z˙ 6 C. BEN´ ETEAU´ AND D. KHAVINSON

The left hand side of equation (2.3) is real and therefore has argument increment 0, while the right hand side has an argument increment of at least 4π as we travel along Γ, unless ϕ0 = 0. (Note that we need to be careful if the expression on the left hand side passes through a zero on ∂Ω.) Hence, ϕ = const and (2.1) implies that Γ = {z : |z − const| = λ}, a disk. 2

Equation (2.1) is closely connected to the “Riccati equation”: since Γ is analytic, near each component γj there is a single-valued branch of an analytic function S(z) (the Schwarz function, see [2, 12, 56] and also Section 5) such thatz ¯ = S(z) on Γ. Then p u := S0(z) is a single-valued analytic function in a tubular neighborhood of ∂Ω, and dz¯ u(z) = on Γ, ds so, after differentiating one more time with respect to the arc-length parameter, (2.1) becomes the Riccati equation (2.4) u2 − iλu0 = f, where f = ϕ0. Since Riccati’s equation is easily transformed into a homogeneous second order linear equation (see [27]) which may only have two linearly indepen- dent solutions, it is yet another indication that if (2.1) holds on Γ, Ω must be at most doubly connected and disks and annuli are the only domains for which (2.1) may hold. Yet Conjecture 2.1 is still open even for doubly connected domains! The Riccati equation (2.4) appears in many free boundary problems, some of which we will discuss in more detail in Section 4. We now turn to the connection with overdetermined boundary problems and Serrin’s theorem.

3. Overdetermined Boundary Value Problems and Serrin’s Theorem

Recall that condition (iii) from Theorem 2.2 states: Z Z 1 1 fdA = fds A Ω P Γ for analytic f in Ω. If Ω is simply connected, this condition is equivalent to: Z Z 1 1 (3.1) udA = uds A Ω P Γ for all functions u harmonic in Ω. Moreover, A. Kosmodem’yansky showed that condition (3.1) is equivalent to the following. Theorem 3.1. ([38]) Consider the solution v of the Dirichlet problem ∆v = 1 in Ω; v = 0 on Γ.

Then the normal derivative of v must satisfy vn = A/P on Γ. THE ISOPERIMETRIC INEQUALITY 7

Indeed, take any harmonic test function u in Ω that is smooth up to the boundary. By Green’s formula, Z Z Z A uvnds = udA = uds. Γ Ω P Γ

Since u is arbitrary, vn = A/P on Γ. In this context, the shape of Ω was already known. We state the following result due to Serrin in two dimensions, although the theorem is more general and holds in all dimensions. Theorem 3.2. ([55]) If the overdetermined boundary value problem ∆v = 1 in Ω, v = 0 on Γ,

vn = const on Γ, has a smooth solution in Ω, then Ω is a disk.

This leads to an equivalent form of Conjecture 2.1 “`ala Serrin” ([34]): Conjecture 3.1. Let Ω be a multiply connected domain. If the overdetermined boundary value problem (n ≥ 2) ∆v = 1 in Ω ∂v A = on Γ ∂n P

v|γj = cj, j = 1, . . . , n, cn = 0 has a smooth solution in Ω, then Ω must be an annulus.

Serrin’s theorem has a natural interpretation in terms of hydrodynamics (see [35, p. 2-3] and [40, p. 653-654]). Suppose we have a viscous, incompressible Newtonian fluid flowing through a pipe with cross-section Ω. A Newtonian fluid is a fluid in which the shear stress, that is, the stress exerted by the fluid that is tangential to the wall of the pipe, is proportional to the velocity gradient in the direction perpendicular to the plane of shear. (We will see shortly in the discussion below why this is true.) The constant of proportionality is known as the viscosity µ. Intuitively, the viscosity measures how “sticky” the fluid is. A viscous fluid is therefore one such that µ 6= 0, like oil or tar. A fluid is incompressible means the density of the fluid is constant along its flow lines; because of the continuity equations in fluid dynamics that express conservation of mass, this implies that the divergence of the velocity vector ~v is zero, i.e., Div(~v) = O · ~v = 0. Now let us assume in addition that the fluid is flowing in lines parallel say to the ~k axis (such a flow is called laminary) and that the flow d~v ~ is steady, i.e., dt = 0. Then the continuity equation implies that ~v = v(x, y)k depends only on x and y. Then the Navier-Stokes equations in this simplified context can be written as 1 ∂p ∆v = − , µ ∂z 8 C. BEN´ ETEAU´ AND D. KHAVINSON where p is the pressure at each point (x, y, z) and depends only on z, since the ∂2p flow is laminary. Since dz2 = 0, the right hand side is actually a constant A, giving us (modulo a constant multiple) the first equation in Serrin’s theorem. On the other hand, the force exerted by the water on the walls of the pipe is given by 4 F~ = (p − µ O · ~v) ~n + µ(~n × (O × ~v)), 3 where ~n is an outward unit normal vector (see [35, 40]). This breaks the force F~ down into a normal component and a tangential, or shear, component. Since ~v = v~k only has a non-zero contribution v in the ~k direction and O · ~v = 0, the force F~ simplifies to: ∂v F~ = p ~n + µ ~k. ∂n ∂v In other words, the quantity µ ∂n represents the “shear stress” on the wall, which is to be expected, as discussed earlier, for a Newtonian fluid. Finally, because we are dealing with a viscous fluid, and the walls of the pipe are fixed (not moving), the condition that v = 0 on Γ is the so-called “adherence” condition. Therefore Serrin’s theorem says that the shear stress at each point on the wall is the same if and only if the cross-section of the pipe is a disk. H. Weinberger ([60]) gave an alternative proof of Serrin’s theorem based on the strong maximum principle for the auxiliary function ϕ := |∆v|2 + v and a Rellich type identity. Serrin and Weinberger’s methods were extended to more general equations and boundary conditions but always with the additional provision that the solution of the overdetermined boundary value problem in question does not achieve local extrema inside the domain. For example, the following result combines the efforts of J. Serrin (’71), G. Alessandrini (’92), W. Reichel (’96), N. Willms, M. Gladwell, and D. Siegel (’94), B. Sirakov (’01), L. Payne and P. Schaeffer (’89), and also G. Philippin and L. Ragoub (’95) (see [41, 42, 43, 44, 45, 46, 47] and references therein). Theorem 3.3. The overdetermined boundary value problem ∆v = 1 ∂v = a ∂n j

v|γj = cj, j = 1, . . . , n, cn = 0, an ≥ 0, cj < 0, aj ≤ 0 has a solution if and only if Ω is a disk or an annulus, and, accordingly, v is a radial function.

However, our isoperimetric problem equivalent to the overdetermined boundary A value problem (Conjecture 3.1) has all aj = P , hence v certainly attains its min- imum inside Ω. Therefore the methods developed to prove results like Theorem 3.3 do not work. THE ISOPERIMETRIC INEQUALITY 9

We now turn to a discussion of condition (ii) in Theorem 2.2 and a related appli- cation to determining the shape of droplets of conducting fluid in the presence of an electric field.

4. Droplets

Recall one of the equivalent conditions for λ(Ω) = 2A/P : z¯(s) − iλz¯˙(s) = ϕ(z(s)) for some ϕ ∈ AΩ. We would like to consider a more general problem, in which the function ϕ may not be continuous in the closure of Ω and may possibly have poles inside Ω. However, we will still need to consider functions that are relatively “good” near the boundary. In order for polynomials or rational functions to be dense in the appropriate space of analytic functions, we will also need to require some a priori regularity of the domain itself. Let us begin therefore with a discussion of Smirnov classes and Smirnov domains. Smirnov classes are one of the two standard generalizations of Hardy classes to arbitrary domains. Let us suppose Ω is a finitely connected domain in C bounded by rectifiable Jordan curves. We say a function f analytic in Ω belongs to the Smirnov class E1 := E1(Ω) if there exists a constant M and an increasing sequence of domains Ωk with boundaries Γk that consist of a finite number of rectifiable Jordan curves such that

∪kΩk = Ω, and Z sup |f(z)| ds ≤ M < ∞. k Γk In this definition, the domains and their boundaries appear to depend on the function, and therefore it is not obvious that E1 is even a linear space. However a theorem of Keldysh and Lavrentiev ([29]) for simply connected domains, ex- tended by S. Ya. Khavinson and Tumarkin ([37]) to finitely connected domains, ensures that this dependence is superfluous. It follows that the function f has non-tangential boundary values f ∗(ζ) almost everywhere on Γ, with f ∗ ∈ L1(Γ). In addition, the Cauchy integral formula holds for f: Z 1 f ∗(ζ) f(z) = dζ 2πi Γ ζ − z for all z ∈ Ω. In fact, the existence of an analytic function’s representation as a Cauchy integral of its non-tangential boundary values can be taken as an equiv- alent definition of its membership in E1, in domains with rectifiable boundaries. ∗ 1 Defining kfkE1 = kf k(L1(Γ),ds) makes E equipped with this norm into a Banach space. For a more complete discussion of Smirnov classes and generalized Hardy spaces, see [13, Chapter 10], [20, Chapter X]. See also the Appendix in [36, p. 57-61]. Now let us consider a conformal mapping ϕ of Ω onto a domain whose boundary consists of finitely many circles or points. Such a conformal mapping always 10 C. BEN´ ETEAU´ AND D. KHAVINSON exists by Koebe’s theorem (see [20, p. 237-238]). Since Ω has a rectifiable boundary Γ, ϕ0 can be shown to be in E1(Ω). We say that Ω is a Smirnov domain if, for each z ∈ Ω, Z 1 ∂g (ζ, z) log |ϕ0(z)| = log |ϕ0(ζ)| Ω |dζ|, 2π Γ ∂n where gΩ(ζ, z) is the Green function of Ω having singularity at z. In other words, ϕ0 has no singular part and the harmonic function log |ϕ0| can be recovered from its boundary values via the Poisson formula. Let us now state a more general problem. Problem 4.1. Find all Smirnov domains Ω ∈ Cˆ := C ∪ ∞, whose boundary Γ = n S 1 γj, consists of n rectifiable Jordan curves, such that there exists F (z) ∈ E (Ω) 1 analytic, or F meromorphic in Ω and in E1 close to the boundary of Ω and with prescribed poles z1, . . . , zk in Ω, such that

F (z) = pjz¯ + iτjz¯˙ + cj on γj, 2 2 for some constants pj, τj ∈ R, pj + τj 6= 0, cj ∈ C. (If ∞ ∈ Ω we will always assume it to be one of the poles.)

This problem turns out to be interesting in determining the shape of electrified droplets of fluid: suppose Γ is the boundary of a planar droplet of perfectly conducting fluid in the presence of an electrostatic field E. Let p be the fluid pressure inside the droplet and τ the surface tension, and call Ω the unbounded component of the droplet. E has a harmonic potential u, that is, E = −Ou, and u is the real part of an analytic potential g. This analytic potential, after various normalizations (see [35, p. 15]), can be written as α α g(z) = z + 1 + 2 + ... z z2 P. Garabedian ([17]) showed that if the droplet is in equilibrium, dz¯ (4.1) F (z) = pz¯ + iτ on Γ, ds where Z 2α (4.2) F (z) = (g0)2dz = z + 1 + ... z for z ∈ Ω.F is called an integrated analytic potential. In [35], every solution to (4.1) on its Jordan boundary Γ that has the form (4.2) for√z ∈ Ω is defined to be a mathematical droplet. For a “physical droplet”, g0 = F 0 must be single valued in Ω. If τ is very large, the p-term in (4.1) is negligible, and the resulting equation also describes small air bubbles in fluid flow ([17, 39]). A lower bound for τ should exist as well. For τ = 0 there are certainly no physical droplets. Yet, mathematical droplets for that case are ellipses ([12, 35, 56]). The following result shows the existence of a one parameter family of mathematical droplets. THE ISOPERIMETRIC INEQUALITY 11

Theorem 4.2. ([35, p. 24-26]) There exists a one parameter family of unbounded domains Ωt, each with rectifiable boundary Γt, and a corresponding family of functions Ft analytic in Ωt except for a simple pole with residue 1 at ∞, such that dz¯ F (z) = p z¯ + iτ on Γ , t t t ds t for some real constants pt and τt, with pt 6= 0, τt 6= 0.

Each of these domains Ωt is thus an example of a solution to Problem 4.1. Their boundaries Γt are images of the unit circle under a rational mapping of degree 3 on which (4.1) holds. None of these curves however is a physical droplet. To our knowledge, no other examples of such domains are known. In particular, we do not know of any examples of transcendental curves satisfying (4.1), although, most likely, there are plenty of them! Applying electrical forces to droplets of conducting fluid has led to some very concrete applications: the process of “electrowetting”, for example, in which an electric force is applied at the interface of a droplet of conducting fluid and a solid, has applications to digital cameras, camera phones, and home security systems. In 2003, scientists from Philips Research created a fluid lens that operates on the basis of the process of electrowetting: two non-mixing fluids, one conducting and one not, are placed inside a tube. The layer between the liquids (the meniscus) acts as a lens. An electric field is applied to the tube, which causes the conducting fluid to change its shape, thus resulting in a change of the focal length of the lens. See [48] for more details. For further references on electrowetting and its applications, see [5, 24]. A slightly different type of application can be found in [11]: there, the authors use Schwarz functions to model the changing shape of a void created and traveling inside a thin metal conductor subjected to an intense electric field. This model is similar in some ways to the one used for Hele-Shaw flows (see [10, 23, 52]).

5. Some special cases

Let us now examine three distinguished cases of Problem 4.1, in which the bound- n ary condition on Γ = ∪j=1γj simplifies to one of the following:

(5.1) F (z) = pjz¯ z ∈ γj, pj ∈ R − {0};

(5.2) F (z) = iτjz¯˙ z ∈ γj, τj ∈ R − {0};

(5.3) F (z) = pjz¯ + cj z ∈ γj, pj ∈ R − {0}, cj ∈ C. 12 C. BEN´ ETEAU´ AND D. KHAVINSON

Note that the existence of a function F satisfying (5.2) implies the existence of a function g satisfying (5.3): simply define Z g(z) = (F (z))2dz.

Then, by (5.2), for z ∈ γj, we have Z Z dz¯ (F (z))2dz = −τ 2 ( )2dz = −τ 2(¯z + c ), j ds j j for some constant cj. Therefore g is well-defined as a single valued analytic func- tion, and (5.3) holds. From now on, we shall always assume additional regularity for Ω, i.e., that Ω is a Jordan Smirnov domain.

5.1. The Schwarz function. Notice that the first special case (5.1) is inti- mately connected with the Schwarz function, since on each boundary component, F/pj is the Schwarz function for γj. More specifically, let us recall the definition of the Schwarz function of a curve (see [12, 56]): Definition. Suppose Γ is a non-singular real-analytic Jordan arc in C. Then there is a neighborhood G of Γ and a uniquely determined analytic function S on G such that S(z) =z ¯ for z ∈ Γ. S is called the Schwarz function of Γ.

(5.1) then means that all of the Schwarz functions F/pj of the curves γj are connected to each other, each one being a real multiple of F. The following simple theorem is stated in [35]: Theorem 5.1. ([35, Thm 5.3]) If Ω is bounded and (5.1) holds with F ∈ E1(Ω), then Ω must be an annulus.

Sketch of proof. Since f ∈ E1(Ω) and Ω is bounded, zF (z) ∈ E1(Ω). Be- 2 cause of the boundary condition (5.1), zF (z) = pj|z| on γj, so zF (z) is also in L∞(Γ). Finally, since Ω is a Smirnov domain, the above two conditions imply that zF (z) ∈ H∞(Ω) (see [13]). But zF (z) is analytic in Ω and real-valued on Γ, and therefore zF (z) is constant. (Notice that the condition that Ω is Smirnov is crucial here.) Therefore |z| is constant on each boundary component, so Ω must be an annulus or a disk centered at the origin. Because F (z) = c/z is not analytic in the disk, Ω must be an annulus. 2

Note that the coefficients p1 and p2 in the above theorem are not arbitrary: they are connected to the radii of the annulus, and are not equal to each other. If we require the coefficients pj to be equal, therefore, there is no analytic function that satisfies (5.1) for a bounded domain Ω. On the other hand, if we allow F to have poles a1, . . . , am, and require the coefficients pj to be equal (without loss of THE ISOPERIMETRIC INEQUALITY 13 generality pj = 1), then Ω is a so-called quadrature domain; namely, if f is any function analytic in Ω, then by the complex form of Green’s theorem, Z Z Z 1 1 Xm fdA = fzdz¯ = fF dz = π f(a )Res F. 2i 2i j aj Ω Γ Γ j=1

These domains have been intensely studied in the 1980s by D. Aharonov, B. Gustafsson, H. S. Shapiro, K. Ullemar, Y. Avsi (see [56] and references therein). Also, see [23] for an account of many recent developments.

Even when we do not require the coefficients pj to be equal, a similar argument as in the proof of Theorem 5.1 shows that if F is assumed to have a simple pole at the origin and Ω is bounded, then Ω must be a disk. (This is well-known in the context of Schwarz functions: the Schwarz function of a domain has one pole if and only if the domain is a disk.) If the function F has two different poles (and if the coefficients pj are different), then the problem is already more difficult.

5.2. Vekua’s Problem. The second special case (5.2)

F (z) = iτjz¯˙ z ∈ γj, τj ∈ R − {0} is a particular example of an overdetermined boundary value problem made enormously popular by works of I. N. Vekua in the 1950s. It is not difficult to see that (5.2) implies (see [59]) that F is orthogonal on Γ to all functions analytic in Ω with a single-valued primitive, hence

Xn−1 ∂ω F (z) = c j , j ∂z j=1 for some real constants cj, where ωj are the harmonic measures of the boundary components γj with respect to Ω, i.e., ωj (j = 1, . . . , n − 1) are harmonic inside Ω and equal to 1 on γj and 0 on Γ − γj. Define Xn−1 u(z) = cjωj(z). j=1 Then u solves the following overdetermined boundary value problem in Ω:  ∆u = 0 (V ) u = cj on γj (j = 1, . . . , n − 1), cn = 0  ∂u ∂n = τj on γj (j = 1, . . . , n). For this special case of Vekua’s overdetermined boundary value problem, Ω is known, in the doubly-connected case: Theorem 5.2. ([35, Thm 5.5]) If Ω is a doubly-connected bounded Smirnov domain, and (V) is solvable, then Ω is an annulus. 14 C. BEN´ ETEAU´ AND D. KHAVINSON

The problem is open for domains of connectivity 3 and higher. The authors of [35] conjecture that there are no solutions in that case. Again, if we assume that all the constants on all boundary components are the same in (5.2), that is, F (z) = iτz,¯˙ τ 6= 0, z ∈ Γ, and we allow F to have poles a1, . . . , am, then for all f analytic in Ω, we have: Z Z 1 2π Xm fds = F fdz = f(a )Res F. iτ τ j aj Γ Γ 1 This type of domain Ω is called an arc-length quadrature domain and much is known about them (see [23, 56]). For example, simply connected arc-length quadrature domains are conformal images of the unit disk under maps by rational functions of a rather special form (see [56, Thm 5.4]). Note that the assump- tion that Ω is Smirnov is crucial, otherwise there exist, for example, one point arc-length quadrature domains, so-called pseudo-circles, with highly non-smooth boundaries (see [29, 50, 57]). There are many applications of quadrature domains to fluid dynamics. One of the most well-known examples is the connection between quadrature domains and Hele-Shaw flows, which was discovered by S. Richardson in [52]. A detailed survey of applications of quadrature domains to fluid dynamics can be found in [10].

5.3. Unbounded domains. Let us mention briefly the case of unbounded (Smirnov) domains Ω, with Jordan boundary Γ. Notice that we are now con- sidering the boundary Γ to consist of a single curve. Recently, P. Jones and S. Smirnov ([28]) showed that if a domain is Smirnov, then its complement is most likely non-Smirnov! This provided an unexpected negative solution to a prob- lem posed by S. Ya. Khavinson in the 50’s, which proposed to find an intrinsic characterization of a Smirnov curve: the theorem of Jones and Smirnov revealed therefore that such a characterization is not possible. If there exists a function F satisfying (5.2), the shape of Γ will depend on the behavior of F at infinity. Let us begin by considering functions analytic at ∞ in Ω. Theorem 5.3. ([14]) Let Γ be a Jordan curve whose exterior Ω is a Smirnov domain. Suppose there exists F ∈ E1(Ω), analytic at ∞, such that dz¯ (5.4) F (z) = a.e. on Γ. ds c Then Γ is a circle and F (z) = z . In [14], the authors assumed that F (∞) = 0; in [35], this hypothesis was shown to be redundant. In potential-theoretic terms, this statement is equivalent to saying that if the equilibrium mass distribution is uniformly distributed on Γ (with respect to arc-length), then Γ must be a circle (see [14] for details). This THE ISOPERIMETRIC INEQUALITY 15 holds in all dimensions provided that Γ is a C2 surface (see [51]). Equivalently, if the overdetermined boundary value problem ∆u = 0 in Ω; u = const 6= 0 on Γ; ∂u = const on Γ ∂n has a solution in an (unbounded!) domain Ω, then Γ is a circle. We may also consider the case where F has a simple pole at infinity. Recall that this context has a physical interpretation, discussed in Section 4, as a droplet of conducting fluid in which the surface tension is much larger than the pressure inside the droplet (which is then considered negligible). In this case, the following theorem gives an example of a family of mathematical droplets.

Theorem 5.4. ([35, Thm 6.2]) Let Γ be a Jordan curve, with√ (logarithmic) 3+2 3 capacity 1, whose exterior Ω is a Smirnov domain. If τ ≥ 3 and there exists F ∈ E1 near the boundary of Ω and with a simple pole at ∞, that is, 1 F = z + O( z ), and dz¯ (5.5) F = iτ on Γ, ds then Γ is included into one parameter family {Γt}, t = 1/τ, where Γt is the image of the unit circle under the conformal mapping 1 t2 ϕ (w) = − 2tw − w3. t w 3 √ 3+2 3 For τ ≤ 3 , (5.5) has no solution among mathematical droplets with Jordan boundaries. The droplets are convex for τ ≥ 3 and the family contains only one physical droplet corresponding to the value τ = 3.

6. Extensions to higher dimensions

Finally, let us discuss what is known in higher dimensions. Suppose Ω is a bounded domain in Rn, n ≥ 3, Γ is the boundary of Ω,V is the volume of Ω, and P is the (surface) area of Γ. Let H(Ω) be the closure in the uniform norm on Ω¯ of the space of functions harmonic in a neighborhood of Ω. More generally, if K is a compact subset of Rn, and C(K) is the space of continuous functions on K, we will write H(K) for the uniform closure in C(K) of the space of functions harmonic in a neighborhood of K. Pn 1 n 2 2 2 Let x = (x1, . . . , xn) be a vector in R , and |x| = ( j=1 xj ) . If one thinks of H(K) as the uniform closure of the kernel of the Laplace operator ∆ and R(K) as the uniform closure of the kernel of the operator ∂/∂z,¯ then the analogy of the anti-analytic functionz ¯ is the function |x|2, since (∂/∂z¯)(¯z) = 1 and ∆(|x|2) = 2n = const 6= 0. With this in mind, we define the concept of harmonic content as follows. 16 C. BEN´ ETEAU´ AND D. KHAVINSON

Definition. The harmonic content of K is defined to be 2 Λ(K) := distC(K)(| x| ,H(K)).

For a bounded domain Ω, we will write Λ(Ω) := Λ(Ω)¯ . We then have the following result. Theorem 6.1. ([32]) Λ(K) = 0 ⇔ H(K) = C(K).

Note that in the case of analytic content in C, the equivalence of the statements λ(K) = 0 and R(K) = C(K) follows at once from the Stone-Weierstrass theorem, since R(K) is an algebra. However, Theorem 6.1 is non-trivial, since H(K) is not an algebra. Different proofs were given by Poletsky ([49]) and Bliedtner (see [7] and references therein, in particular to the works of W. Hansen).

Harmonic content can be estimated in terms of geometric quantities. If Rharm ¯ is the radius of the ball with the same capacity as Ω, and Rvol is the radius of the ball with the same volume as Ω, then the following theorem gives upper and lower bounds for the harmonic content of a domain Ω. Theorem 6.2. ([32, 33]) 1 1 R2 ≤ Λ(Ω) ≤ R2 2 harm 2 vol and equality on either side occurs only for balls.

The upper estimate was proved in [32], and the lower estimate as well as exten- sions of both inequalities to general elliptic operators were obtained in [33]. An interesting extension of this result to approximation in C1-norm by harmonic functions is due to Gauthier and Paramonov (see [19]). Note that the harmonic content Λ(Ω) leads to a different isoperimetric inequality, Rharm ≤ Rvol. A lower bound depending only on volume and perimeter cannot occur, because one can construct a “Swiss cheese” set K ⊂ R2 that has positive area and finite perimeter, yet so that H(K) = C(K), that is, such that Λ(K) = 0 (see [25]). In order to get a better hold of the simple geometric quantities of a domain, let us now consider another analogue of the concept of analytic content in higher dimensions. Recall that analytic content for a domain Ω in C is defined as

λ(Ω) := inf k z¯ − ϕ kC(Ω)¯ . ϕ∈AΩ Note that this is also equal to

λ(Ω) := inf k z − ϕ¯ kC(Ω)¯ . ϕ∈AΩ

An anti-analytic functionϕ ¯ = f1 +if2 can be identified with the harmonic vector field f = (f1, f2) = ∇u, u a harmonic real-valued function, where −→ −→ div f = curl f = 0. THE ISOPERIMETRIC INEQUALITY 17

(See the discussion in [22, p. 76].) This motivates our definition of analytic content in higher dimensions: define the space A(Ω) of harmonic vector fields −→ −→ 1 ¯ f = (f1, . . . , fd) in Ω as the set of all vector fields f ∈ C (Ω) ∩ C(Ω) such that −→ −→ div f = curl f = 0. We define −→ B(Ω) = {∇ h , h ∈ H(Ω)¯ }. Unless Ω is simply-connected, B(Ω) A(Ω). n −→ P 2 1/2 Also k f k∞ = supx∈Ω( fj (x)) . Then we define the analytic content λ(Ω) of 1 a domain in Rn as follows. Definition. −→ −→ λ(Ω) = dist( x , A(Ω)) := inf k x − fk∞. f∈A(Ω)

The analogue of Theorem 2.1 is then the following:

Theorem 6.3. ([22]) There exists a constant cn > 0 such that nV (Ω) ≤ λ(Ω) ≤ c V 1/n(Ω). P (∂Ω) n The lower bound is sharp since equality occurs for balls and spherical shells (i.e., n a set of the form {x ∈ R : r < |x − x0| < R}).

The constant cn was calculated explicitly in [22] and is equal to n1+1/nΓ(n/2)Γ( 2n−1 )1−1/n 2n−2 , 2n−1 n2 2n 1−1/n 2π Γ( 2n−2 ) where, here, Γ is the usual Gamma function.p In the case n = 2, the quantity 1/n cnV (Ω) reduces to the sharp estimate Area(Ω)/π. The obstacle in proving the sharpness of the upper bound for n ≥ 3 comes from the fact that in the Ahlfors-Beurling estimate for the maximum Z dV (y) max k∇( n−2 )k∞, x∈Ω Ω |x − y| the extremal solids are not balls in all dimensions ≥ 3, although they are very symmetric algebraic surfaces that are getting more and more tightly sealed to the tangent plane at the maximum point (see [22, p. 82]). The following conjecture proposed in [22] remains open.

Conjecture 6.1. λ(Ω) ≤ Rvol.

The following theorem is the analogue of Theorem 2.2 and gives conditions equiv- alent to the attainment of the lower bound in Theorem 6.3. 18 C. BEN´ ETEAU´ AND D. KHAVINSON

Theorem 6.4. ([22])TFAE: nV (i) λ(Ω) = P .

(ii) There exists −→ϕ ∈ B(Ω)(!) : −→x − λ−→n (x) = −→ϕ (x) on ∂Ω, where −→n is the outward unit normal to ∂Ω. R R R 1 1 ∂u (iii) V Ω u dV = P ∂Ω u dσ for all u harmonic in Ω such that S ∂n dσ = 0 for all closed surfaces S in Ω.

(iv) There exists u in Ω satisfying ∆u = 1; ∂u | = const; ∂n ∂Ω u|∂Ω = local constant.

The following conjecture thus follows naturally: nV Conjecture 6.2. λ(Ω) = P ⇔ Ω is either a ball or a spherical shell. Serrin’s theorem in higher dimensions implies that if an extremal domain is home- omorphic to a ball, then it must be a ball; however Conjecture 6.2 is still open for domains whose boundary contains more than one component, or domains (such as a torus) that are not homeomorphic to a ball.

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Department of Mathematics and Computer Science, Seton Hall University, South Orange , New Jersey 07079 E-mail address: [email protected]

Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701 E-mail address: [email protected]

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